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Highly Connected Sets and the Excluded Grid Theorem

Reinhard Diestel, Tommy R. Jensen, Konstantin Yu. Gorbunov, Carsten Thomassen
1999 Journal of combinatorial theory. Series B (Print)  
We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width.  ...  We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of vertices.  ...  Highly connected sets and tangles The proofs of Theorem 2 given in 7] and 3] rely heavily on a concept central to the Robertson-Seymour theory of minors but not so far considered in this paper, the concept  ... 
doi:10.1006/jctb.1998.1862 fatcat:urjg4vgcq5ep3nlxaq36ekfweu

Contraction Bidimensionality: The Accurate Picture [chapter]

Fedor V. Fomin, Petr Golovach, Dimitrios M. Thilikos
2009 Lecture Notes in Computer Science  
We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth.  ...  As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory-the meta algorithmic framework to design efficient parameterized and approximation  ...  Finally, we show the following analogue of the Excluding Grid-minor Theorem.  ... 
doi:10.1007/978-3-642-04128-0_63 fatcat:viiyykcgtzf5lbiurgekwtwesa

TOWARDS A MATROID-MINOR STRUCTURE THEORY [chapter]

Jim Geelen, Bert Gerards, Geoff Whittle
2007 Combinatorics, Complexity, and Chance  
In combination with the grid theorem this yields the following theorem.  ...  Theorem 5. 5 ( 5 Grid theorem for matroids) Let H be a planar graph and q be a prime power.  ... 
doi:10.1093/acprof:oso/9780198571278.003.0005 fatcat:rnrqxwsn6fdhdixknsck3bxqqy

The Structure of W_4-Immersion-Free Graphs [article]

Rémy Belmonte and Archontia Giannopoulou and Daniel Lokshtanov and Dimitrios M. Thilikos
2016 arXiv   pre-print
We study the structure of graphs that do not contain the wheel on 5 vertices W4 as an immersion, and show that these graphs can be constructed via 1, 2, and 3-edge-sums from subcubic graphs and graphs  ...  Lemma 2 essentially states that large treewidth yields a large number of vertex disjoint cycles that are highly connected to each other, and an additional disjoint set that is highly connected to these  ...  Other major results in graph minor theory include the (Strong) Structure Theorem [28] , the Weak Structure Theorem [27] , the Excluded Grid Theorem [26, 31, 21] , as well as numerous others, e.g.,  ... 
arXiv:1602.02002v1 fatcat:ewc656g3grgtpbpkn3oedndpki

Some Recent Progress and Applications in Graph Minor Theory

Ken-ichi Kawarabayashi, Bojan Mohar
2007 Graphs and Combinatorics  
In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the "rough" structure of graphs excluding a fixed minor.  ...  This result was used to prove Wagner's Conjecture that finite graphs are well-quasi-ordered under the graph minor relation.  ...  Theorems 5.3 and 5.4 are sharp in the sense that the 7-connectivity and 9-connectivity (respectively) conditions cannot be relaxed, as mentioned above.  ... 
doi:10.1007/s00373-006-0684-x fatcat:wkf3w6cemzc4hnnw3neqpck6hu

Page 46 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
(RS-INT2; Moscow); Thomassen, Carsten (DK-TUD; Lyngby) Highly connected sets and the excluded grid theorem. (English summary) J. Combin. Theory Ser. B 75 (1999), no. 1, 61-73.  ...  Summary: “We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width.  ... 

On the block number of graphs [article]

Daniel Weißauer
2017 arXiv   pre-print
We also study k-blocks in graphs from classes of graphs G that exclude some fixed graph as a topological minor, and prove that every G ∈G satisfies β(G) ≤ c√(|G|) for some constant c = c( G).  ...  The block number β(G) of G is the maximum integer k for which G contains a k-block.  ...  of Theorem 3.  ... 
arXiv:1702.04245v1 fatcat:n56svml2mzgftji5es3rqhuv3u

Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask)

Dániel Marx, Michal Pilipczuk, Marc Herbstritt
2014 Symposium on Theoretical Aspects of Computer Science  
On the algorithmic side, our study reveals for example that an unexpected combination of bounded degree, genus, and feedback vertex set number of G gives rise to a highly nontrivial algorithm for Subgraph  ...  We show that all the questions arising in this framework are answered by a set of 11 maximal positive results (algorithms) and a set of 17 maximal negative results (hardness proofs); some of these results  ...  As Theorems 2 and 3 show, |V (H)| is a highly relevant parameter for the problem.  ... 
doi:10.4230/lipics.stacs.2014.542 dblp:conf/stacs/MarxP14 fatcat:dy4trfjwkvhj3i2tzcvrguedyi

Graph Minors and Parameterized Algorithm Design [chapter]

Dimitrios M. Thilikos
2012 Lecture Notes in Computer Science  
We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex  ...  We present some of the basic algorithmic techniques and methods that emerged from this theory.  ...  Acknowledgements We wish to thank Isolde Adler, Marcin Kamiński and Stavros G. Kolliopoulos for their detailed comments, remarks, and suggestions on this text.  ... 
doi:10.1007/978-3-642-30891-8_13 fatcat:56shhf7tevfdpczn4m7qir3zti

Layered separators in minor-closed graph classes with applications

Vida Dujmović, Pat Morin, David R. Wood
2017 Journal of combinatorial theory. Series B (Print)  
Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with n^O(1)n volume, whereas the previous best bound was O(n^3/2).  ...  We extend these results with a O( n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a ^O(1)n bound on the queue-number of graphs excluding a fixed minor.  ...  Thanks to Zdeněk Dvořák, Gwenaël Joret, Sergey Norin, Bruce Reed and Paul Seymour for helpful discussions. Thanks to the anonymous referees for numerous helpful comments.  ... 
doi:10.1016/j.jctb.2017.05.006 fatcat:gzpqanrjr5ebddcetqpoaa6rvm

The Complexity of Conjunctive Queries with Degree 2 [article]

Matthias Lanzinger
2022 arXiv   pre-print
Using dilutions we observe an analogue to the Excluded Grid Theorem for degree 2 hypergraphs.  ...  We also generalise our main structural result to arbitrary bounded degree and discuss possible paths towards a characterisation of the bounded degree case.  ...  Grohe's lower bound critically relies on the Excluded Grid Theorem by Robertson and Seymour [27] .  ... 
arXiv:2111.11532v3 fatcat:lxuylqth3zfgjfpf2lex4szuha

Distribution grid topology reconstruction: An information theoretic approach

Yizheng Liao, Yang Weng, Meng Wu, Ram Rajagopal
2015 2015 North American Power Symposium (NAPS)  
One of the key issues is frequent distribution grid re-configuration, which is hard to detect based on traditional approaches.  ...  Wrong topology information causes wrong control signal, making fast changing smart grid prone to go over stability boundaries and to collapse.  ...  ACKNOWLEDGMENT The second author would like to thank Tie Liu from Texas A&M University for discussions on the Chow-Liu algorithm.  ... 
doi:10.1109/naps.2015.7335248 fatcat:gcyiipid6vfrxl7wxqmmwh5oc4

Recent work in matroid representation theory

Geoff Whittle
2005 Discrete Mathematics  
This paper surveys recent work in matroid representation theory and discusses a number of open problems.  ...  Sets of fields: excluded minors The deepest part of Theorem 3.1 is the excluded-minor characterization of regular matroids.  ...  Firstly there are no analogues to the Wheels and Whirls Theorem and the Splitter Theorem for vertically 4-connected matroids.  ... 
doi:10.1016/j.disc.2004.07.039 fatcat:it6qr2fri5gntjitdyoa7fk43m

Extremal infinite graph theory

Maya Stein
2011 Discrete Mathematics  
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree.  ...  Acknowledgement The author was supported by Fondecyt grant no. 11090141.  ...  We use the vertex-/edge-degree of the ends (for a definition see below) to force highly connected subgraphs and grid minors.  ... 
doi:10.1016/j.disc.2010.12.018 fatcat:4enetufd25ddbhke2ga4qham2a

On the tree-width of even-hole-free graphs [article]

Pierre Aboulker, Isolde Adler, Eun Jung Kim, Ni Luh Dewi Sintiari, Nicolas Trotignon
2020 arXiv   pre-print
This can be seen as a strengthening of Robertson and Seymour's excluded grid theorem for the case of minor-free graphs.  ...  Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width.  ...  Even-hole-free graphs excluding a minor In this section we prove an 'induced grid theorem' for graphs excluding a fixed minor.  ... 
arXiv:2008.05504v1 fatcat:ij2smflzl5dx5h6sladobes5ve
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